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Full flexibility of the Monge-Ampère system in codimension $d_*-d+1$

Wentao Cao, Jonas Hirsch, Dominik Inauen, Marta Lewicka

Abstract

We prove that $\mathcal{C}^{1,α}$ solutions to the Monge-Ampère system in dimension $d$ and codimension $k= d_*-d+1$, where $d_*$ denotes the Janet dimension, are dense in the space of continuous functions, for every Hölder exponent $α<1$. Our result strengthens the statement in [Lewicka 2022], obtained for $k = 2d_*$ and based on ideas from [Källen 1978] in the context of the isometric immersion system. It also generalizes the result of [Inauen-Lewicka 2025], where full flexibility was established in dimension $d=2$ and codimension $k=2$. The same proof scheme further yields local full flexibility of isometric immersions of $d$-dimensional Riemannian metrics into Euclidean space of dimension $d_* + 1$, generalizing the result in [Lewicka 2025] proved for $d=k=2$. By using techniques of [Conti-De Lellis-Szekelyhidi], the result can be extended to compact manifolds, in codimension $(d+1)d_*-d+1$.

Full flexibility of the Monge-Ampère system in codimension $d_*-d+1$

Abstract

We prove that solutions to the Monge-Ampère system in dimension and codimension , where denotes the Janet dimension, are dense in the space of continuous functions, for every Hölder exponent . Our result strengthens the statement in [Lewicka 2022], obtained for and based on ideas from [Källen 1978] in the context of the isometric immersion system. It also generalizes the result of [Inauen-Lewicka 2025], where full flexibility was established in dimension and codimension . The same proof scheme further yields local full flexibility of isometric immersions of -dimensional Riemannian metrics into Euclidean space of dimension , generalizing the result in [Lewicka 2025] proved for . By using techniques of [Conti-De Lellis-Szekelyhidi], the result can be extended to compact manifolds, in codimension .

Paper Structure

This paper contains 8 sections, 11 theorems, 140 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Comparison with the literature
  3. Overview of the strategy of proofs
  4. Organization of the paper and notation
  5. The preparatory statements
  6. The stage construction and a proof of Theorem \ref{['thm_stage']}
  7. The Nash-Kuiper scheme and a proof of Theorem \ref{['th_final']}
  8. Extension to the isometric immersion system

Key Result

Theorem 1.1

Let $\omega\subset\mathbb{R}^d$ be an open, bounded, $d$-dimensional set. Let the target codimension: Given vector fields $v\in\mathcal{C}^1(\bar{\omega},\mathbb{R}^{\bar{d}})$, $w\in\mathcal{C}^1(\bar{\omega},\mathbb{R}^{d})$ and a matrix field $A\in\mathcal{C}^{r,\beta}(\bar{\omega},\mathbb{R}^{d\times d}_\mathrm{sym})$, assume: in the sense of matrix inequalities. Then, for every exponent $\a

Figures (1)

  • Figure 1: Progression of frequencies, intermediate fields, defects and codimensions used in the proof of Theorem \ref{['thm_stage']}.

Theorems & Definitions (12)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Theorem 4.1
  • ...and 2 more